Some hyperbolic three-manifolds that bound geometrically
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- by Alexander Kolpakov, Bruno Martelli and Steven Tschantz PDF
- Proc. Amer. Math. Soc. 143 (2015), 4103-4111 Request permission
Erratum: Proc. Amer. Math. Soc. 144 (2016), 3647-3648.
Abstract:
A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension.
We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and 120-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k\geqslant 1$, we build an orientable compact closed 3-manifold tessellated by $16k$ right-angled dodecahedra that bounds a 4-manifold tessellated by $32k$ right-angled 120-cells.
A notable feature of this family is that the ratio between the volumes of the 4-manifolds and their boundary components is constant and, in particular, bounded.
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Additional Information
- Alexander Kolpakov
- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto Ontario, M5S 2E4, Canada
- MR Author ID: 774696
- Email: kolpakov.alexander@gmail.com
- Bruno Martelli
- Affiliation: Dipartimento di Matematica “Tonelli”, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy
- Email: martelli@dm.unipi.it
- Steven Tschantz
- Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240
- MR Author ID: 174820
- Email: steven.tschantz@vanderbilt.edu
- Received by editor(s): November 12, 2013
- Received by editor(s) in revised form: April 3, 2014, and April 4, 2014
- Published electronically: April 6, 2015
- Additional Notes: The first author was supported by the SNSF researcher scholarship P300P2-151316.
The second author was supported by the Italian FIRB project “Geometry and topology of low-dimensional manifolds”, RBFR10GHHH - Communicated by: Kevin Whyte
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4103-4111
- MSC (2010): Primary 57N16; Secondary 52B11, 52C45
- DOI: https://doi.org/10.1090/proc/12520
- MathSciNet review: 3359598